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- W2892198733 abstract "In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group $Gamma$ with respect to a sequence of groups $left{G_{n}right}_{n=1}^{infty}$, equipped with bi-invariant metrics $left{d_{n}right}_{n=1}^{infty}$. We consider the case $G_{n}=operatorname{U}left(nright)$ (resp. $G_{n}=operatorname{Sym}left(nright)$), equipped with the normalized Hilbert-Schmidt metric $d_{n}^{operatorname{HS}}$ (resp. the normalized Hamming metric $d_{n}^{operatorname{Hamming}}$). Our main result is that if $Gamma$ is infinite, hyperlinear (resp. sofic) and has Property $operatorname{(T)}$, then it is not stable with respect to $left(operatorname{U}left(nright),d_{n}^{operatorname{HS}}right)$ (resp. $left(operatorname{Sym}left(nright),d_{n}^{operatorname{Hamming}}right)$). This answers a question of Hadwin and Shulman regarding the stability of $operatorname{SL}_{3}left(mathbb{Z}right)$. We also deduce that the mapping class group $operatorname{MCG}left(gright)$, $ggeq 3$, and $operatorname{Aut}left(mathbb{F}_nright)$, $ngeq 3$, are not stable with respect to $left(operatorname{Sym}left(nright),d_{n}^{operatorname{Hamming}}right)$. Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on $operatorname{U}left(nright)$ and the (unnormalized) $p$-Schatten metrics, since many groups with Property $operatorname{(T)}$ are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to $left(operatorname{U}left(nright),d_{n}^{operatorname{HS}}right)$ and $left(operatorname{Sym}left(nright),d_{n}^{operatorname{Hamming}}right)$." @default.
- W2892198733 created "2018-09-27" @default.
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- W2892198733 date "2018-09-03" @default.
- W2892198733 modified "2023-10-17" @default.
- W2892198733 title "Group stability and Property (T)" @default.
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